In the operation of automatic grinding machines there are a number of problems that arise which limit production rate and quality of the produced workpieces. One problem is that of "rounding-up" irregular or eccentric stock on machines of low rigidity where the grinding wheel tends to follow original workpiece runout errors. As the workpiece moves by the grinding wheel, variations in the depth of the stock cause the wheel to deflect. This causes the grinding force to vary, resulting in different rates of grinding and a slow rounding-up action results. These problems are evident in cylindrical grinding, particularly when the wheel or workpiece is not rigidly supported. Furthermore, some workpieces exhibit a threshold force below which no cutting takes place, so that rounding-up action ceases if the grinding force drops below this value. These and other difficulties experienced with the prior art devices have been obviated in a novel manner by the present invention.